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Intersection Theory in Differential Algebraic Geometry: Generic Intersections and the Differential Chow Form
Algebraic Geometry · Submitted 1 Sep 2010
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim As a consequence, the dimension conjecture for generic differential polynomials is proved.
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The abstract, sentence by sentence
In this paper, an intersection theory for generic differential polynomials is presented. The intersection of an irreducible differential variety of dimension $d$ and order $h$ with a generic differential hypersurface of order $s$ is shown to be an irreducible variety of dimension $d-1$ and order $h+s$. As a consequence, the dimension conjecture for generic differential polynomials is proved. Based on the intersection theory, the Chow form for an irreducible differential variety is defined and most of the properties of the Chow form in the algebraic case are established for its differential counterpart. Furthermore, the generalized differential Chow form is defined and its properties are proved. As an application of the generalized differential Chow form, the differential resultant of $n+1$ generic differential polynomials in $n$ variables is defined and properties similar to that of the Macaulay resultant for multivariate polynomials are proved.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper, an intersection theory for generic differential polynomials is presented. | contribution 1.00 | 0.93 | 0.10 | asserted 1.00 |
| s1 | The intersection of an irreducible differential variety of dimension $d$ and order $h$ with a generic differential hypersurface of order $s$ is shown to be an irreducible variety of dimension $d-1$ and order $h+s$. | result 1.00 | 0.90 | 0.38 | asserted 1.00 |
| s2 | As a consequence, the dimension conjecture for generic differential polynomials is proved. | result 0.85 | 0.91 | 0.28 | asserted 1.00 |
| s3 | Based on the intersection theory, the Chow form for an irreducible differential variety is defined and most of the properties of the Chow form in the algebraic case are established for its differential counterpart. | contribution 0.99 | 0.93 | 0.15 | asserted 1.00 |
| s4 | Furthermore, the generalized differential Chow form is defined and its properties are proved. | contribution 0.98 | 0.94 | 0.09 | asserted 1.00 |
| s5 | As an application of the generalized differential Chow form, the differential resultant of $n+1$ generic differential polynomials in $n$ variables is defined and properties similar to that of the Macaulay resultant for multivariate polynomials are proved. | contribution 0.86 | 0.93 | 0.14 | asserted 1.00 |
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The paper
Original abstract
In this paper, an intersection theory for generic differential polynomials is presented. The intersection of an irreducible differential variety of dimension $d$ and order $h$ with a generic differential hypersurface of order $s$ is shown to be an irreducible variety of dimension $d-1$ and order $h+s$. As a consequence, the dimension conjecture for generic differential polynomials is proved. Based on the intersection theory, the Chow form for an irreducible differential variety is defined and most of the properties of the Chow form in the algebraic case are established for its differential counterpart. Furthermore, the generalized differential Chow form is defined and its properties are proved. As an application of the generalized differential Chow form, the differential resultant of $n+1$ generic differential polynomials in $n$ variables is defined and properties similar to that of the Macaulay resultant for multivariate polynomials are proved.
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How the literature uses it
4 arXiv papers cite this in 20 sentences
Papers citing it · 16 sentences without a confident label
Cites it as background 3 papers
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Meanwhile, its counterpart the differential resultant is at an initial state of development, a survey on this development can be found in the introductions of [ref] and [ref] .
Linear sparse differential resultant formulas · Introduction · 0.99
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In [ref] , the first rigorous definition for the differential resultant of $n+1$ differential polynomials in $n$ differential indeterminates was given and its properties were proved.
Matrix Formula of Differential Resultant for First Order Generic Ordinary Differential Polynomials · Introduction · 0.87
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For more details about differential algebra, please refer to [ref] .
Sparse Differential Resultant for Laurent Differential Polynomials · Preliminaries · 1.00
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